Yet when Mordechai later asked her to approach the king to save all the Jews, she resisted and said she couldn't approach him without having been summoned.

What happened?

I can think of a couple plausible explanations, but I don't know if any of them is correct:

Esther just happened to be summoned before the king in the first instance. (But if so, why didn't the Megillah tell us this, since in the later instance it tells us just how dangerous it is to go without being summoned - wouldn't this be yet another instance of hidden Divine assistance?)

Esther was admonished for approaching the king without having been summoned in the first instance. (This could explain her rather lengthy explanation later, not to mention her emphasis that, "everyone in the kingdom knows that you can't do this.")

A twist to the second scenario above: the first instance resulted in a change in the law (and/or a widely publicized announcement of the law).

It was simply due to the nature of the assignment: the first being to save the king, the second being to save herself and her people.

Mightn't the first case have been accomplished with a note or a messenger? Or, perhaps she was being called to the king regularly at the time, but explicitly wasn't ("these thirty days") later on.
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Isaac MosesFeb 25 '13 at 20:09

Esther 4:11 explicitly gives the reason why Esther was nervous. "but I have not been summoned to come to the king these thirty days."

She felt the king was not warm to her at the time, and that made her nervous; seemingly that was not the case at the time of her first visit, either because she had then been summoned within the previous thirty days or because the king had not then been acting coolly toward her.

The Persian kings had a well-established process—a secret police, even—for reporting threats against the crown. (Source: Rabbi Yehuda Landy’s Purim and the Persian Empire, quoting historical sources.) Esther may have used those channels for reporting Bigthan and Teresh rather than approaching the king directly.

(Personal speculation: This may also explain why Mordechai was not rewarded immediately: although his name was entered into the record [since “Esther told the king”—or his agents—“in Mordechai’s name”], initial credit for the report would have been given to Esther.)